# Purely horizontal forcesF1 and F2 are applied in such

1 Beam bendingA uniformly thick beam of length L with square cross-section with width a has a distributed force,N/m,applied to (Q is a constant). A point force P is applied at .Choose the beam dimension a to reduce the deflection to below 1 cm whilst minimising the mass of the beam. Sketch distributions of shear forces, moment and deflection along the beam length.You are expected (but not required) to use MatLab to solve this problem.Values of L, E, Q and P for each group are calculated as follows:(here N is group ID number) N= 18(m)(kN)(kN/m)(GPa)Marking scheme:Finding correct expressions for shear and moment ANALYTICALLY (not using Matlab, please show your working) (8 marks)Finding correct expression for deflection (analytically or using Matlab) (8 marks)Finding the optimum beam dimensions (analytically or using Matlab) (6 marks)Plots of shear force, momentum and deflection (3 marks)2 Car suspensionThe car suspension helps to reduce the amount of bounce when car is going over a bump. Cars drive on roads that have bumps or holes on them – large ones (sleeping policemen), medium sized ones (potholes) or even small sinusoidal bumps.Performance of car suspension is often evaluated using a complex system of springs and dampers, e.g. (Shirahatt et al, J. of the Braz. Soc. of Mech. Sci. & Eng. 2008):We will use a simplified model of a quarter car: one wheel and its suspension that carries a quarter of weight of a car. The wheel axle is connected to the remaining quarter car by a spring and a shock absorber (damper):The mass of the car is M, the mass of the wheel is m, the height of the car body is y(t) above the level h = 0 (not above the road surface!), the height of the wheel axel above h = 0 is s(t) and the length of the string in the “unloaded” state is l.We make the following assumptions:(1) Purely horizontal forcesF1 and F2 are applied in such a way as to maintain the quarter car vertical and travelling forward at a constant speed v m/s.(2) Wheel tyre behaviour is not modelled so we assume that the size of the wheel is constant and hence where r is the radius of the wheel.(3) The car is moving in equilibrium when x = 0 and t = 0 (i.e. y is not changing before t=0)(4) We ignore mass of the spring and the damperThe car is travelling over a series of bumps described by:The task is:(a) Write down equations of motion with the correct forms for the spring force and damper force and set correct initial conditions (8 marks)(b) Solve the equations for the height of the car y(t) on the road with the above bumps(8 marks)(c) Calculate the vertical acceleration and the vertical jerk () experienced by the passengers. (5 marks)(d) For given values of suspension length, spring and damper constants produce graphs of height of the car body, the vertical acceleration and the vertical jerk experienced by the passengers for a car driving 5, 10, 15 and 30 miles per hour. (8 marks)(e) For the above range of speeds, plot the maximum height of the car body, the maximum magnitude of vertical acceleration and the maximum magnitude of vertical jerk versus speed (ignore the initial acceleration/jerk when starting to go over the bumps). Interpret the results. In particular, at what speed or speeds is going over the speed bumps most violent? (8 marks)(f) Maximum jerk of 18 m/s3 is often used as a criterion for a comfortable ride. Which of the above speeds provide a comfortable ride? (4 marks)(g) For extra points: if you wanted to reduce the jerk, which parameters of the suspension system (spring constant, spring length, damper constant) would you change and how? Justify your answer and attach a plot with an example of improved jerk for new parameters.(4 marks)You are expected to use Matlab for solving the ODE and plotting.Reference values are:Bump height(m)Bump length(m)Wheel radius(cm)Suspension (spring) length(cm)Wheel mass15 kgValues of spring constant k for each group are calculated as follows (here N is group ID number):(N/m)Values of quarter-car mass Mand damper constant b for each group are taken from the table:Quarter car mass M®500 (kg)550 (kg)600 (kg)650 (kg)Damper constant b¯3000 (N s/m)N = 1-4N = 5-8N = 9-12N = 13-164500 (N s/m)N = 17-20N = 21-24N = 25-28N = 29-326000 (N s/m)N = 33-36N = 37-40N = 41-44N = 45-487500 (N s/m)N = 49-52N = 53-56N = 57-60N = 61-64(for example, group number 39 is in the third row and second column therefore M = 550 kg and b = 6000 Ns/m)3 Normal modesThree masses are connected by springs on a smooth horizontal table and can move in a straight line. The force of the weight of each mass in, say, the y direction is matched by the reaction of the table. The only resultant forces on the masses come from the springs.The first mass is connected to a wall by another spring. Assume the motions are sufficiently small so that the springs remain linear.The task is:(a) Write down the equations of motion for the three masses. (10 marks)(b) Determine the frequency of the normal modes of the system and the normal modes of motion. (10 marks)(c) At t = 0 we displace all three masses from the equilibrium by 0.01 m and then let go. Plot the displacement for all three masses vs time. (5 marks)(d) For extra points: if the system were to be stimulated by a small time varying force, applied to the last mass, where w is close to one of the normal mode frequencies, explain what outcome you would expect to see for the output of the system. If possible produce a Mupad notebook to illustrate the behaviour you would expect(5 marks)The values of k1 – k3 and m1 – m3 are calculated as follows (N is your group ID number):(factor to be used for calculations below)(N/m), (N/m), (N/m)(kg), (kg), (kg)